src/HOL/Lifting.thy
author kuncar
Thu Feb 20 16:56:33 2014 +0100 (2014-02-20)
changeset 55610 9066b603dff6
parent 55604 42e4e8c2e8dc
child 55731 66df76dd2640
permissions -rw-r--r--
refactoring; generate rep_eq always, not only when it would be accepted by the code generator
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(*  Title:      HOL/Lifting.thy
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    Author:     Brian Huffman and Ondrej Kuncar
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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header {* Lifting package *}
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theory Lifting
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imports Equiv_Relations Transfer
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keywords
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  "parametric" and
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  "print_quot_maps" "print_quotients" :: diag and
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  "lift_definition" :: thy_goal and
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  "setup_lifting" "lifting_forget" "lifting_update" :: thy_decl
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begin
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subsection {* Function map *}
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context
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begin
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interpretation lifting_syntax .
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lemma map_fun_id:
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  "(id ---> id) = id"
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  by (simp add: fun_eq_iff)
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subsection {* Other predicates on relations *}
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definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
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lemma left_totalI:
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  "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
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unfolding left_total_def by blast
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lemma left_totalE:
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  assumes "left_total R"
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  obtains "(\<And>x. \<exists>y. R x y)"
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using assms unfolding left_total_def by blast
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lemma bi_total_iff: "bi_total A = (right_total A \<and> left_total A)"
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unfolding left_total_def right_total_def bi_total_def by blast
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lemma bi_total_conv_left_right: "bi_total R \<longleftrightarrow> left_total R \<and> right_total R"
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by(simp add: left_total_def right_total_def bi_total_def)
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definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
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lemma left_unique_transfer [transfer_rule]:
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  assumes [transfer_rule]: "right_total A"
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  assumes [transfer_rule]: "right_total B"
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  assumes [transfer_rule]: "bi_unique A"
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  shows "((A ===> B ===> op=) ===> implies) left_unique left_unique"
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using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def fun_rel_def
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by metis
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lemma bi_unique_iff: "bi_unique A = (right_unique A \<and> left_unique A)"
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unfolding left_unique_def right_unique_def bi_unique_def by blast
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lemma bi_unique_conv_left_right: "bi_unique R \<longleftrightarrow> left_unique R \<and> right_unique R"
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by(auto simp add: left_unique_def right_unique_def bi_unique_def)
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lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
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unfolding left_unique_def by blast
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lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
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unfolding left_unique_def by blast
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lemma left_total_fun:
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  "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
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  unfolding left_total_def fun_rel_def
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  apply (rule allI, rename_tac f)
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  apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI)
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  apply clarify
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  apply (subgoal_tac "(THE x. A x y) = x", simp)
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  apply (rule someI_ex)
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  apply (simp)
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  apply (rule the_equality)
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  apply assumption
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  apply (simp add: left_unique_def)
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  done
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lemma left_unique_fun:
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  "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
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  unfolding left_total_def left_unique_def fun_rel_def
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  by (clarify, rule ext, fast)
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lemma left_total_eq: "left_total op=" unfolding left_total_def by blast
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lemma left_unique_eq: "left_unique op=" unfolding left_unique_def by blast
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lemma [simp]:
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  shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
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  and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
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by(auto simp add: left_unique_def right_unique_def)
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lemma [simp]:
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  shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
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  and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
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by(simp_all add: left_total_def right_total_def)
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subsection {* Quotient Predicate *}
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definition
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  "Quotient R Abs Rep T \<longleftrightarrow>
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     (\<forall>a. Abs (Rep a) = a) \<and> 
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     (\<forall>a. R (Rep a) (Rep a)) \<and>
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     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
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     T = (\<lambda>x y. R x x \<and> Abs x = y)"
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lemma QuotientI:
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  assumes "\<And>a. Abs (Rep a) = a"
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    and "\<And>a. R (Rep a) (Rep a)"
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    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
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    and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
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  shows "Quotient R Abs Rep T"
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  using assms unfolding Quotient_def by blast
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context
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  fixes R Abs Rep T
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  assumes a: "Quotient R Abs Rep T"
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begin
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lemma Quotient_abs_rep: "Abs (Rep a) = a"
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  using a unfolding Quotient_def
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  by simp
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lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel:
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  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
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  using a unfolding Quotient_def
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  by fast
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lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
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  using a unfolding Quotient_def
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  by fast
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lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
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  using a unfolding Quotient_def
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  by metis
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lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rep_abs_fold_unmap: 
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  assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 
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  shows "R (Rep' x') x"
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proof -
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  have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
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  then show ?thesis using assms(3) by simp
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qed
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lemma Quotient_Rep_eq:
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  assumes "x' \<equiv> Abs x" 
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  shows "Rep x' \<equiv> Rep x'"
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by simp
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lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel_abs2:
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  assumes "R (Rep x) y"
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  shows "x = Abs y"
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proof -
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  from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
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  then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
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qed
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lemma Quotient_symp: "symp R"
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  using a unfolding Quotient_def using sympI by (metis (full_types))
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lemma Quotient_transp: "transp R"
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  using a unfolding Quotient_def using transpI by (metis (full_types))
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lemma Quotient_part_equivp: "part_equivp R"
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by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
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end
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lemma identity_quotient: "Quotient (op =) id id (op =)"
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unfolding Quotient_def by simp 
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text {* TODO: Use one of these alternatives as the real definition. *}
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lemma Quotient_alt_def:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
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    (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
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apply safe
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (simp (no_asm_use) only: Quotient_def, fast)
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apply (rule QuotientI)
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apply simp
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apply metis
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apply simp
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apply (rule ext, rule ext, metis)
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done
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lemma Quotient_alt_def2:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
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    (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
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  unfolding Quotient_alt_def by (safe, metis+)
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lemma Quotient_alt_def3:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
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    (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
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  unfolding Quotient_alt_def2 by (safe, metis+)
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lemma Quotient_alt_def4:
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  "Quotient R Abs Rep T \<longleftrightarrow>
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    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
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  unfolding Quotient_alt_def3 fun_eq_iff by auto
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lemma fun_quotient:
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  assumes 1: "Quotient R1 abs1 rep1 T1"
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  assumes 2: "Quotient R2 abs2 rep2 T2"
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  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
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  using assms unfolding Quotient_alt_def2
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  unfolding fun_rel_def fun_eq_iff map_fun_apply
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  by (safe, metis+)
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lemma apply_rsp:
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  fixes f g::"'a \<Rightarrow> 'c"
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  assumes q: "Quotient R1 Abs1 Rep1 T1"
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  and     a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rsp':
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  assumes a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rsp'':
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  assumes "Quotient R Abs Rep T"
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  and "(R ===> S) f f"
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  shows "S (f (Rep x)) (f (Rep x))"
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proof -
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  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
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  then show ?thesis using assms(2) by (auto intro: apply_rsp')
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qed
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subsection {* Quotient composition *}
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lemma Quotient_compose:
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  assumes 1: "Quotient R1 Abs1 Rep1 T1"
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  assumes 2: "Quotient R2 Abs2 Rep2 T2"
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  shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
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  using assms unfolding Quotient_alt_def4 by fastforce
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lemma equivp_reflp2:
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  "equivp R \<Longrightarrow> reflp R"
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  by (erule equivpE)
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subsection {* Respects predicate *}
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definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
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  where "Respects R = {x. R x x}"
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lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
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  unfolding Respects_def by simp
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subsection {* Invariant *}
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definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
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  where "invariant R = (\<lambda>x y. R x \<and> x = y)"
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lemma invariant_to_eq:
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  assumes "invariant P x y"
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  shows "x = y"
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using assms by (simp add: invariant_def)
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lemma fun_rel_eq_invariant:
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  shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
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by (auto simp add: invariant_def fun_rel_def)
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lemma invariant_same_args:
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  shows "invariant P x x \<equiv> P x"
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using assms by (auto simp add: invariant_def)
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lemma invariant_transfer [transfer_rule]:
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  assumes [transfer_rule]: "bi_unique A"
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  shows "((A ===> op=) ===> A ===> A ===> op=) Lifting.invariant Lifting.invariant"
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unfolding invariant_def[abs_def] by transfer_prover
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lemma UNIV_typedef_to_Quotient:
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  assumes "type_definition Rep Abs UNIV"
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  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
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  shows "Quotient (op =) Abs Rep T"
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proof -
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  interpret type_definition Rep Abs UNIV by fact
kuncar@47361
   318
  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
kuncar@47361
   319
    by (fastforce intro!: QuotientI fun_eq_iff)
kuncar@47308
   320
qed
kuncar@47308
   321
kuncar@47361
   322
lemma UNIV_typedef_to_equivp:
kuncar@47308
   323
  fixes Abs :: "'a \<Rightarrow> 'b"
kuncar@47308
   324
  and Rep :: "'b \<Rightarrow> 'a"
kuncar@47308
   325
  assumes "type_definition Rep Abs (UNIV::'a set)"
kuncar@47308
   326
  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
kuncar@47308
   327
by (rule identity_equivp)
kuncar@47308
   328
huffman@47354
   329
lemma typedef_to_Quotient:
kuncar@47361
   330
  assumes "type_definition Rep Abs S"
kuncar@47361
   331
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
kuncar@47501
   332
  shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
kuncar@47361
   333
proof -
kuncar@47361
   334
  interpret type_definition Rep Abs S by fact
kuncar@47361
   335
  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
kuncar@47361
   336
    by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
kuncar@47361
   337
qed
kuncar@47361
   338
kuncar@47361
   339
lemma typedef_to_part_equivp:
kuncar@47361
   340
  assumes "type_definition Rep Abs S"
kuncar@47501
   341
  shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
kuncar@47361
   342
proof (intro part_equivpI)
kuncar@47361
   343
  interpret type_definition Rep Abs S by fact
kuncar@47501
   344
  show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
kuncar@47361
   345
next
kuncar@47501
   346
  show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
kuncar@47361
   347
next
kuncar@47501
   348
  show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
kuncar@47361
   349
qed
kuncar@47361
   350
kuncar@47361
   351
lemma open_typedef_to_Quotient:
kuncar@47308
   352
  assumes "type_definition Rep Abs {x. P x}"
huffman@47354
   353
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
kuncar@47308
   354
  shows "Quotient (invariant P) Abs Rep T"
huffman@47651
   355
  using typedef_to_Quotient [OF assms] by simp
kuncar@47308
   356
kuncar@47361
   357
lemma open_typedef_to_part_equivp:
kuncar@47308
   358
  assumes "type_definition Rep Abs {x. P x}"
kuncar@47308
   359
  shows "part_equivp (invariant P)"
huffman@47651
   360
  using typedef_to_part_equivp [OF assms] by simp
kuncar@47308
   361
huffman@47376
   362
text {* Generating transfer rules for quotients. *}
huffman@47376
   363
huffman@47537
   364
context
huffman@47537
   365
  fixes R Abs Rep T
huffman@47537
   366
  assumes 1: "Quotient R Abs Rep T"
huffman@47537
   367
begin
huffman@47376
   368
huffman@47537
   369
lemma Quotient_right_unique: "right_unique T"
huffman@47537
   370
  using 1 unfolding Quotient_alt_def right_unique_def by metis
huffman@47537
   371
huffman@47537
   372
lemma Quotient_right_total: "right_total T"
huffman@47537
   373
  using 1 unfolding Quotient_alt_def right_total_def by metis
huffman@47537
   374
huffman@47537
   375
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
huffman@47537
   376
  using 1 unfolding Quotient_alt_def fun_rel_def by simp
huffman@47376
   377
huffman@47538
   378
lemma Quotient_abs_induct:
huffman@47538
   379
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
huffman@47538
   380
  using 1 assms unfolding Quotient_def by metis
huffman@47538
   381
huffman@47537
   382
end
huffman@47537
   383
huffman@47537
   384
text {* Generating transfer rules for total quotients. *}
huffman@47376
   385
huffman@47537
   386
context
huffman@47537
   387
  fixes R Abs Rep T
huffman@47537
   388
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
huffman@47537
   389
begin
huffman@47376
   390
huffman@47537
   391
lemma Quotient_bi_total: "bi_total T"
huffman@47537
   392
  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
huffman@47537
   393
huffman@47537
   394
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
huffman@47537
   395
  using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
huffman@47537
   396
huffman@47575
   397
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
huffman@47575
   398
  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
huffman@47575
   399
huffman@47889
   400
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
huffman@47889
   401
  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
huffman@47889
   402
huffman@47537
   403
end
huffman@47376
   404
huffman@47368
   405
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
huffman@47368
   406
huffman@47534
   407
context
huffman@47534
   408
  fixes Rep Abs A T
huffman@47368
   409
  assumes type: "type_definition Rep Abs A"
huffman@47534
   410
  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
huffman@47534
   411
begin
huffman@47534
   412
kuncar@51994
   413
lemma typedef_left_unique: "left_unique T"
kuncar@51994
   414
  unfolding left_unique_def T_def
kuncar@51994
   415
  by (simp add: type_definition.Rep_inject [OF type])
kuncar@51994
   416
huffman@47534
   417
lemma typedef_bi_unique: "bi_unique T"
huffman@47368
   418
  unfolding bi_unique_def T_def
huffman@47368
   419
  by (simp add: type_definition.Rep_inject [OF type])
huffman@47368
   420
kuncar@51374
   421
(* the following two theorems are here only for convinience *)
kuncar@51374
   422
kuncar@51374
   423
lemma typedef_right_unique: "right_unique T"
kuncar@51374
   424
  using T_def type Quotient_right_unique typedef_to_Quotient 
kuncar@51374
   425
  by blast
kuncar@51374
   426
kuncar@51374
   427
lemma typedef_right_total: "right_total T"
kuncar@51374
   428
  using T_def type Quotient_right_total typedef_to_Quotient 
kuncar@51374
   429
  by blast
kuncar@51374
   430
huffman@47535
   431
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
huffman@47535
   432
  unfolding fun_rel_def T_def by simp
huffman@47535
   433
huffman@47534
   434
end
huffman@47534
   435
huffman@47368
   436
text {* Generating the correspondence rule for a constant defined with
huffman@47368
   437
  @{text "lift_definition"}. *}
huffman@47368
   438
huffman@47351
   439
lemma Quotient_to_transfer:
huffman@47351
   440
  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
huffman@47351
   441
  shows "T c c'"
huffman@47351
   442
  using assms by (auto dest: Quotient_cr_rel)
huffman@47351
   443
kuncar@47982
   444
text {* Proving reflexivity *}
kuncar@47982
   445
kuncar@47982
   446
lemma Quotient_to_left_total:
kuncar@47982
   447
  assumes q: "Quotient R Abs Rep T"
kuncar@47982
   448
  and r_R: "reflp R"
kuncar@47982
   449
  shows "left_total T"
kuncar@47982
   450
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
kuncar@47982
   451
kuncar@55563
   452
lemma Quotient_composition_ge_eq:
kuncar@55563
   453
  assumes "left_total T"
kuncar@55563
   454
  assumes "R \<ge> op="
kuncar@55563
   455
  shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="
kuncar@55563
   456
using assms unfolding left_total_def by fast
kuncar@51994
   457
kuncar@55563
   458
lemma Quotient_composition_le_eq:
kuncar@55563
   459
  assumes "left_unique T"
kuncar@55563
   460
  assumes "R \<le> op="
kuncar@55563
   461
  shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="
noschinl@55604
   462
using assms unfolding left_unique_def by blast
kuncar@47982
   463
kuncar@52307
   464
lemma left_total_composition: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
kuncar@52307
   465
unfolding left_total_def OO_def by fast
kuncar@52307
   466
kuncar@52307
   467
lemma left_unique_composition: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
noschinl@55604
   468
unfolding left_unique_def OO_def by blast
kuncar@52307
   469
kuncar@55563
   470
lemma invariant_le_eq:
kuncar@55563
   471
  "invariant P \<le> op=" unfolding invariant_def by blast
kuncar@55563
   472
kuncar@55563
   473
lemma reflp_ge_eq:
kuncar@55563
   474
  "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast
kuncar@55563
   475
kuncar@55563
   476
lemma ge_eq_refl:
kuncar@55563
   477
  "R \<ge> op= \<Longrightarrow> R x x" by blast
kuncar@47982
   478
kuncar@51374
   479
text {* Proving a parametrized correspondence relation *}
kuncar@51374
   480
kuncar@51374
   481
definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
kuncar@51374
   482
"POS A B \<equiv> A \<le> B"
kuncar@51374
   483
kuncar@51374
   484
definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
kuncar@51374
   485
"NEG A B \<equiv> B \<le> A"
kuncar@51374
   486
kuncar@51374
   487
(*
kuncar@51374
   488
  The following two rules are here because we don't have any proper
kuncar@51374
   489
  left-unique ant left-total relations. Left-unique and left-total
kuncar@51374
   490
  assumptions show up in distributivity rules for the function type.
kuncar@51374
   491
*)
kuncar@51374
   492
kuncar@51374
   493
lemma bi_unique_left_unique[transfer_rule]: "bi_unique R \<Longrightarrow> left_unique R"
kuncar@51374
   494
unfolding bi_unique_def left_unique_def by blast
kuncar@51374
   495
kuncar@51374
   496
lemma bi_total_left_total[transfer_rule]: "bi_total R \<Longrightarrow> left_total R"
kuncar@51374
   497
unfolding bi_total_def left_total_def by blast
kuncar@51374
   498
kuncar@51374
   499
lemma pos_OO_eq:
kuncar@51374
   500
  shows "POS (A OO op=) A"
kuncar@51374
   501
unfolding POS_def OO_def by blast
kuncar@51374
   502
kuncar@51374
   503
lemma pos_eq_OO:
kuncar@51374
   504
  shows "POS (op= OO A) A"
kuncar@51374
   505
unfolding POS_def OO_def by blast
kuncar@51374
   506
kuncar@51374
   507
lemma neg_OO_eq:
kuncar@51374
   508
  shows "NEG (A OO op=) A"
kuncar@51374
   509
unfolding NEG_def OO_def by auto
kuncar@51374
   510
kuncar@51374
   511
lemma neg_eq_OO:
kuncar@51374
   512
  shows "NEG (op= OO A) A"
kuncar@51374
   513
unfolding NEG_def OO_def by blast
kuncar@51374
   514
kuncar@51374
   515
lemma POS_trans:
kuncar@51374
   516
  assumes "POS A B"
kuncar@51374
   517
  assumes "POS B C"
kuncar@51374
   518
  shows "POS A C"
kuncar@51374
   519
using assms unfolding POS_def by auto
kuncar@51374
   520
kuncar@51374
   521
lemma NEG_trans:
kuncar@51374
   522
  assumes "NEG A B"
kuncar@51374
   523
  assumes "NEG B C"
kuncar@51374
   524
  shows "NEG A C"
kuncar@51374
   525
using assms unfolding NEG_def by auto
kuncar@51374
   526
kuncar@51374
   527
lemma POS_NEG:
kuncar@51374
   528
  "POS A B \<equiv> NEG B A"
kuncar@51374
   529
  unfolding POS_def NEG_def by auto
kuncar@51374
   530
kuncar@51374
   531
lemma NEG_POS:
kuncar@51374
   532
  "NEG A B \<equiv> POS B A"
kuncar@51374
   533
  unfolding POS_def NEG_def by auto
kuncar@51374
   534
kuncar@51374
   535
lemma POS_pcr_rule:
kuncar@51374
   536
  assumes "POS (A OO B) C"
kuncar@51374
   537
  shows "POS (A OO B OO X) (C OO X)"
kuncar@51374
   538
using assms unfolding POS_def OO_def by blast
kuncar@51374
   539
kuncar@51374
   540
lemma NEG_pcr_rule:
kuncar@51374
   541
  assumes "NEG (A OO B) C"
kuncar@51374
   542
  shows "NEG (A OO B OO X) (C OO X)"
kuncar@51374
   543
using assms unfolding NEG_def OO_def by blast
kuncar@51374
   544
kuncar@51374
   545
lemma POS_apply:
kuncar@51374
   546
  assumes "POS R R'"
kuncar@51374
   547
  assumes "R f g"
kuncar@51374
   548
  shows "R' f g"
kuncar@51374
   549
using assms unfolding POS_def by auto
kuncar@51374
   550
kuncar@51374
   551
text {* Proving a parametrized correspondence relation *}
kuncar@51374
   552
kuncar@51374
   553
lemma fun_mono:
kuncar@51374
   554
  assumes "A \<ge> C"
kuncar@51374
   555
  assumes "B \<le> D"
kuncar@51374
   556
  shows   "(A ===> B) \<le> (C ===> D)"
kuncar@51374
   557
using assms unfolding fun_rel_def by blast
kuncar@51374
   558
kuncar@51374
   559
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
kuncar@51374
   560
unfolding OO_def fun_rel_def by blast
kuncar@51374
   561
kuncar@51374
   562
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
kuncar@51374
   563
unfolding right_unique_def left_total_def by blast
kuncar@51374
   564
kuncar@51374
   565
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
kuncar@51374
   566
unfolding left_unique_def right_total_def by blast
kuncar@51374
   567
kuncar@51374
   568
lemma neg_fun_distr1:
kuncar@51374
   569
assumes 1: "left_unique R" "right_total R"
kuncar@51374
   570
assumes 2: "right_unique R'" "left_total R'"
kuncar@51374
   571
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
kuncar@51374
   572
  using functional_relation[OF 2] functional_converse_relation[OF 1]
kuncar@51374
   573
  unfolding fun_rel_def OO_def
kuncar@51374
   574
  apply clarify
kuncar@51374
   575
  apply (subst all_comm)
kuncar@51374
   576
  apply (subst all_conj_distrib[symmetric])
kuncar@51374
   577
  apply (intro choice)
kuncar@51374
   578
  by metis
kuncar@51374
   579
kuncar@51374
   580
lemma neg_fun_distr2:
kuncar@51374
   581
assumes 1: "right_unique R'" "left_total R'"
kuncar@51374
   582
assumes 2: "left_unique S'" "right_total S'"
kuncar@51374
   583
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
kuncar@51374
   584
  using functional_converse_relation[OF 2] functional_relation[OF 1]
kuncar@51374
   585
  unfolding fun_rel_def OO_def
kuncar@51374
   586
  apply clarify
kuncar@51374
   587
  apply (subst all_comm)
kuncar@51374
   588
  apply (subst all_conj_distrib[symmetric])
kuncar@51374
   589
  apply (intro choice)
kuncar@51374
   590
  by metis
kuncar@51374
   591
kuncar@51956
   592
subsection {* Domains *}
kuncar@51956
   593
kuncar@51956
   594
lemma pcr_Domainp_par_left_total:
kuncar@51956
   595
  assumes "Domainp B = P"
kuncar@51956
   596
  assumes "left_total A"
kuncar@51956
   597
  assumes "(A ===> op=) P' P"
kuncar@51956
   598
  shows "Domainp (A OO B) = P'"
kuncar@51956
   599
using assms
kuncar@51956
   600
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def fun_rel_def 
kuncar@51956
   601
by (fast intro: fun_eq_iff)
kuncar@51956
   602
kuncar@51956
   603
lemma pcr_Domainp_par:
kuncar@51956
   604
assumes "Domainp B = P2"
kuncar@51956
   605
assumes "Domainp A = P1"
kuncar@51956
   606
assumes "(A ===> op=) P2' P2"
kuncar@51956
   607
shows "Domainp (A OO B) = (inf P1 P2')"
kuncar@51956
   608
using assms unfolding fun_rel_def Domainp_iff[abs_def] OO_def
kuncar@51956
   609
by (fast intro: fun_eq_iff)
kuncar@51956
   610
kuncar@53151
   611
definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
kuncar@51956
   612
where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
kuncar@51956
   613
kuncar@51956
   614
lemma pcr_Domainp:
kuncar@51956
   615
assumes "Domainp B = P"
kuncar@53151
   616
shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
kuncar@53151
   617
using assms by blast
kuncar@51956
   618
kuncar@51956
   619
lemma pcr_Domainp_total:
kuncar@51956
   620
  assumes "bi_total B"
kuncar@51956
   621
  assumes "Domainp A = P"
kuncar@51956
   622
  shows "Domainp (A OO B) = P"
kuncar@51956
   623
using assms unfolding bi_total_def 
kuncar@51956
   624
by fast
kuncar@51956
   625
kuncar@51956
   626
lemma Quotient_to_Domainp:
kuncar@51956
   627
  assumes "Quotient R Abs Rep T"
kuncar@51956
   628
  shows "Domainp T = (\<lambda>x. R x x)"  
kuncar@51956
   629
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
kuncar@51956
   630
kuncar@51956
   631
lemma invariant_to_Domainp:
kuncar@51956
   632
  assumes "Quotient (Lifting.invariant P) Abs Rep T"
kuncar@51956
   633
  shows "Domainp T = P"
kuncar@51956
   634
by (simp add: invariant_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
kuncar@51956
   635
kuncar@53011
   636
end
kuncar@53011
   637
kuncar@47308
   638
subsection {* ML setup *}
kuncar@47308
   639
wenzelm@48891
   640
ML_file "Tools/Lifting/lifting_util.ML"
kuncar@47308
   641
wenzelm@48891
   642
ML_file "Tools/Lifting/lifting_info.ML"
kuncar@47308
   643
setup Lifting_Info.setup
kuncar@47308
   644
kuncar@51994
   645
lemmas [reflexivity_rule] = 
kuncar@55563
   646
  order_refl[of "op="] invariant_le_eq Quotient_composition_le_eq
kuncar@55563
   647
  Quotient_composition_ge_eq
kuncar@55563
   648
  left_total_eq left_unique_eq left_total_composition left_unique_composition
kuncar@55563
   649
  left_total_fun left_unique_fun
kuncar@51374
   650
kuncar@51374
   651
(* setup for the function type *)
kuncar@47777
   652
declare fun_quotient[quot_map]
kuncar@51374
   653
declare fun_mono[relator_mono]
kuncar@51374
   654
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
kuncar@47308
   655
wenzelm@48891
   656
ML_file "Tools/Lifting/lifting_term.ML"
kuncar@47308
   657
wenzelm@48891
   658
ML_file "Tools/Lifting/lifting_def.ML"
kuncar@47308
   659
wenzelm@48891
   660
ML_file "Tools/Lifting/lifting_setup.ML"
kuncar@47308
   661
kuncar@55563
   662
hide_const (open) invariant POS NEG
kuncar@47308
   663
kuncar@47308
   664
end